Abstract (Invited)

Elastic Waves in Inhomogeneous Media and Structures

N.S.Bakhvalov, K.Yu.Bogachev (Department of Computational Mathematics, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia); M.E.Eglit (Department of Fluid Mechanics, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia)

e-mail: bakh@abs.math.msu.su

Elastic wave propagation in periodically stratified media as well as in thin layered plates and rods is studied. The ratio d of the length scale of inhomogeneity, or the thickness of the plate and the rod, to the typical wavelength is supposed to be small. The so-called effective equations are derived by the method of two-scale asymptotic expansions [1] over d. The aim is to describe dispersion of waves. That is why the equations up to terms of the second order over d are considered. These terms contain the third and the fourth derivatives of the averaged displacements over time and space coordinates. It is shown that, in contrast to the situation in homogeneous isotropic materials, for inhomogeneous structures, in general, the terms with the third derivatives do enter the equations [2, 3] with the skew-symmetric matrix of the coefficients. This matrix is equal to zero in some cases, e.g., for waves in media consisting of a repeated system of two homogeneous layers with arbitrary local anisotropy; but it differs from zero, e.g., for a two-layer plate even if the layers are homogeneous and isotropic. The dependence of the wave velocity on its length is studied for structures with different types of symmetry. It is shown, in particular, that typically at least one of possible waves displays negative dispersion for every direction of wave propagation. The work was supported by the Russian Foundation of Basic Research (projects 02-01-490, 02-01-613). References: [1]. Bakhvalov N.S., Panasenko G.P., 1984. Homogenization. Averaging Processes in Periodic Media. Mathematical Problems in Mechanics of Composite Materials. Nauka, Moscow. [2]. Bakhvalov N.S., Eglit M.E., 2000. Effective equations with dispersion for wave propagation in periodic media. Doklady RAN, vol. 370, No 1. [3]. Bakhvalov N.S., Eglit M.E., 2002. Investigation of the effective equations with dispersion for wave propagation in stratified media and thin plates. Doklady RAN, vol. 383, No 6.

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